A code $C$ in the Hamming graph $\varGamma=H(m,q)$ is $2$-neighbour-transitive if ${\rm Aut}(C)$ acts transitively on each of $C=C_0$, $C_1$ and $C_2$, the first three parts of the distance partition of $V\varGamma$ with respect to $C$. Previous classifications of families of $2$-neighbour-transitive codes leave only those with an affine action on the alphabet to be investigated. Here, $2$-neighbour-transitive codes with minimum distance at least $5$ and that contain ``small'' subcodes as blocks of imprimitivity are classified. When considering codes with minimum distance at least $5$, completely transitive codes are a proper subclass of $2$-neighbour-transitive codes. This leads, as a corollary of the main result, to a solution of a problem posed by Giudici in 1998 on completely transitive codes.
@article{10_37236_8040,
author = {Neil I. Gillespie and Daniel R. Hawtin and Cheryl E. Praeger},
title = {2-neighbour-transitive codes with small blocks of imprimitivity},
journal = {The electronic journal of combinatorics},
year = {2020},
volume = {27},
number = {1},
doi = {10.37236/8040},
zbl = {1475.94190},
url = {http://geodesic.mathdoc.fr/articles/10.37236/8040/}
}
TY - JOUR
AU - Neil I. Gillespie
AU - Daniel R. Hawtin
AU - Cheryl E. Praeger
TI - 2-neighbour-transitive codes with small blocks of imprimitivity
JO - The electronic journal of combinatorics
PY - 2020
VL - 27
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/8040/
DO - 10.37236/8040
ID - 10_37236_8040
ER -
%0 Journal Article
%A Neil I. Gillespie
%A Daniel R. Hawtin
%A Cheryl E. Praeger
%T 2-neighbour-transitive codes with small blocks of imprimitivity
%J The electronic journal of combinatorics
%D 2020
%V 27
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/8040/
%R 10.37236/8040
%F 10_37236_8040
Neil I. Gillespie; Daniel R. Hawtin; Cheryl E. Praeger. 2-neighbour-transitive codes with small blocks of imprimitivity. The electronic journal of combinatorics, Tome 27 (2020) no. 1. doi: 10.37236/8040
